Sen. Thad Cochran of Mississippi and his allies are adopting a more-pointed message and are looking to make other course corrections to overcome political odds and win a June 24 runoff against a tea party-backed challenger for the Republican Senate nomination.

Mr. Cochran, a six-term incumbent, was slightly outpolled in Tuesday's balloting but wound up in a runoff against state Sen. Chris McDaniel because neither got more than 50%. Now, the Cochran campaign is adopting a sharper tone, saying Mr. McDaniel's fiscal...